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17
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5answers
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Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$

Background: Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$ Starting from $F = ma$, integrating with respect to distance, ...
15
votes
1answer
2k views

What is the definition of how to count degrees of freedom?

This question resulted, rather as by-product, the discussion on how to count degrees of freedom (DOF). I extend that question here: Are necessary1 derivatives such as velocities counted as ...
11
votes
2answers
619 views

In counting degrees of freedom of a linear molecule, why is rotation about the axis not counted?

I was reading about the equipartition theorem and I got the following quotations from my books: A diatomic molecule like oxygen can rotate about two different axes. But rotation about the axis ...
9
votes
1answer
1k views

Gauge fixing and degrees of freedom

Today, my friend (@Will) posed a very intriguing question - Consider a complex scalar field theory with a $U(1)$ gauge field $(A_\mu, \phi, \phi^*)$. The idea of gauge freedom is that two solutions ...
8
votes
1answer
590 views

Clarification on “central charge equals number of degrees of freedom”

It's often stated that the central charge c of a CFT counts the degrees of freedom: it adds up when stacking different fields, decreases as you integrate out UV dof from one fixed point to another, ...
7
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1answer
1k views

Counting degrees of freedom for gravitational waves as a gauge field

Sean Carroll has a new popularization about the Higgs, The Particle at the End of the Universe. Carroll is a relativist, and I enjoyed seeing how he presented the four forces of nature synoptically, ...
7
votes
2answers
452 views

Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics

How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics. I need an clear argument that involves the ...
7
votes
3answers
4k views

What are the six degrees of freedom of the atoms in a solid?

A monoatomic ideal gas has heat capacity $C_v=1.5$ which comes from the three translational degrees of freedom. For solids at high temperature, $C_v=3$, implying six degrees of freedom. What are ...
7
votes
1answer
151 views

Gauge fixing of an arbitrary field

How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
6
votes
6answers
2k views

Degree of freedom paradox for a rigid body

Suppose we consider a rigid body, which has $N$ particles. Then the number of degrees of freedom is $3N - (\mbox{# of constraints})$. As the distance between any two points in a rigid body is fixed, ...
6
votes
1answer
100 views

7/2 versus 9/2 for diatomic heat capacity

Question I calculated the classical heat capacity of a diatomic gas as $C_V = (9/2)Nk_B$, however the accepted value is $C_V = (7/2)Nk_B$. I assumed the classical Hamiltonian of two identical atoms ...
6
votes
2answers
581 views

WHY does the “order” of a differential equation = number of “energy storage” elements in a system?

OK. in all engineering courses there comes a point when they introduce you to systems theory and modeling of systems (for eg. via the impulse response) and then the Laplace transform. The modern ...
5
votes
2answers
833 views

Why energy at room temperature $= kT$ and not $(3/2)kT$ [duplicate]

I always see that a room temperature of $T=300\,\text{K}$ corresponds to an energy of $k_BT \approx \frac{1}{40}\,\text{eV}$. But shouldn't it be $\frac{3}{2}k_BT$ since the molecules in the air have ...
5
votes
2answers
253 views

Extra vibrational mode in linear molecule

When calculating the number of vibrational modes for a molecule, the formulas differ for linear $(n = 3N - 5)$ and non-linear $(n = 3N - 6)$ molecules, where $n$ is number of modes and $N$ is number ...
5
votes
2answers
344 views

Questions about the degree of freedom in General Relatity

I'm confused about the number of degrees of freedom in General Relatity. There are two ways to count it. However, they are contradictory. For simplicity, we consider vacuum solution. First, ...
5
votes
4answers
1k views

Degrees of freedom of the graviton versus classical degrees of freedom

I have a puzzle I can not even understand. A graviton is generally understood in $D$ dimensions as a field with some independent components or degrees of freedom (DOF), from a traceless symmetric ...
4
votes
2answers
151 views

degree of freedom in 6 dimensional space

Let us assume a cartesian space, where the directions are given by $\hat{i},\hat{j},\hat{k}$. The degree of freedom of a rigid body is $6$. The first three correspond to the position coordinates ...
4
votes
2answers
381 views

How many physical degrees of freedom does the $\mathrm{SU(N)}$ Yang-Mills theory have?

The $\mathrm{U(1)}$ QED case has two physical degrees of freedom, which is easy to understand because the free electromagnetic field must be transverse to the direction of propagation. But what are ...
4
votes
2answers
663 views

Uniqueness of the number of degrees of freedom

As per my knowledge, degrees of freedom of any physical system are the number of independent quantities(coordinates) which need to be specified in order to specify the state of a system uniquely. ...
4
votes
2answers
903 views

Counting degrees of freedom in presence of constraints

In a $N$ dimensional phase space if I have $M$ 1st class and $S$ 2nd class constraints, then I have $N-2M-S$ degrees of freedom in phase space. How can I calculate the degrees of freedom in ...
4
votes
3answers
732 views

Why do we say linear molecules only have 2 rotational degrees of freedom? Why does the third 'frozen' one not count?

It is possible to excite rotations around the axes perpendicular to the bond of a linear molecule. However, rotation around the axis along the bond of the molecule would require huge energies, due to ...
4
votes
1answer
186 views

How exactly are the degrees of freedom seen by a falling into a black hole observer related to the ones seen by a staying outside observer?

This is some kind of a follow up of this nicely to the point answer to a provocative (but nevertheless upvoted!) question, about the legitimacy of black hole physics. The answer mentions, that the ...
3
votes
2answers
427 views

$E=kT$ or $\frac32kT$?

Basically, which is the correct formula for thermal energy, and is this the same as kinetic energy? My notes are pretty conflicting on this topic, and I'm getting pretty confused.
3
votes
2answers
187 views

Dark matter: degrees of freedom

I'm afraid this question could sound a little too vague. I don't even know if dark matter (DM) can be genuinely described by quantum field theory, or if quantum field theory should be somehow ...
3
votes
2answers
120 views

Independent components in a 4-vector representing massless fields

In Ryder Page141, it is written "the electromagnetic field, like any massless field, possesses only two independent components, but is covariantly described by a 4-vector $A_{\mu}$". Why are there ...
3
votes
3answers
1k views

Counting degrees of freedom of gauge bosons

Gauge bosons are represented by $A_{\mu}$, where $\mu = 0,1,2,3$. So in general there are 4 degrees of freedom. But in reality, a photon (gauge boson) has two degrees of freedom (two polarization ...
3
votes
2answers
2k views

Polarization vectors of massive and massless particles

I read from Mandl & Shaw that when quantizing massless vector particles such as photons in Lorentz gauge, there are 4 linearly independent polarization vectors (2 of them being able to "gauged ...
3
votes
2answers
8k views

Modeling a two-mass, spring, damper system

I'm trying to model a system with two masses, two springs, two dampers, and one applied force using transfer functions. I'll then be inputting it into Simulink. The system looks like this but there ...
3
votes
1answer
422 views

Propagating degrees of freedom of graviton

What is the best way to see that the number of propagating degrees of freedom or gravitons in 3 dimensions is $0$ ? By graviton I mean the metric and NOT some topologically massive graviton that one ...
3
votes
1answer
119 views

Do photons have six degrees of freedom?

Calculations involving pressure and volume relationships of photon gas during the cosmologic expansion of the universe posit an adiabatic cooling process with a heat capacity ration of 4/3. This ratio ...
3
votes
1answer
438 views

Number $g(T)$ of relativistic degrees of freedom as a function of temperature $T$

Let us consider the total number of relativistic degrees of freedom $g(T)$ for particle species in our universe: $$g(T)=\left(\sum_Bg_B\right)+\frac{7}{8}\left(\sum_Fg_F\right)$$ Where the sums are ...
3
votes
1answer
245 views

Is the ground state of a QFT always a pure state? And excited states are mixed?

I am studying entanglement entropy. It's fullfilled for any local quantum system that the entanglement entropy of a region $A$ in a highly mixed state is extensvie, $$ S_A \sim ...
3
votes
1answer
460 views

Pauli-Fierz “massive” equation and linearized gravity

It it known that the massive spin-2 irreducible representation of the Poincare group is the traceless symmetrical transverse 4-tensor $h_{\mu \nu}$ with rank 2: $$ (\partial^{2} + m^{2})h_{\mu \nu} = ...
3
votes
1answer
327 views

A different proof for 6 degrees of freedom

I want a different proof of 6 degrees of freedom of a solid object made of $N$ particles. I am thinking along these lines: The definition of rigid body is $$\left\lvert \vec{r_i}-\vec{r_j} ...
3
votes
1answer
291 views

Phase Space dimension of Lorenz Strange Attractor

It is often discussed in 3 spatial dimensions and the need for third dimension to prevent self intersection is mentioned. But shouldn't the phase space of the Lorenz system be 6 dimensional, i.e., the ...
3
votes
1answer
98 views

Rigorous definition of degrees of freedom

According to this Wikipedia article, the definition of degrees of freedom is: The degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its ...
3
votes
0answers
208 views

Quantum vs classical degrees of freedom

It is sometimes stated that any classical underpinnings (rightly non-local) of a general quantum system are unrealistic or unphysical because these require exponentially more information to store what ...
3
votes
0answers
76 views

What is the deepest cause of the such high specific heat capacity of water?

Yes, I know about the hydrogen bridges. But I think, it isn't the deepest cause. Anyway, they are only second-order bindings, although quite strong. I think, somehow should have the water a ...
2
votes
3answers
305 views

Natural units of information

In physics entropy is usually measured in nats. I wonder is there a possible model of a physical system which has entropy of discrete number of nats? How particles and degrees of freedom should be ...
2
votes
2answers
3k views

degree of freedom of a rigid body 5 or 6?

I'm confused here. I have a three particle (rigid) system. What would be the degree of freedom? I found out five. 3 coordinates for center of mass and 2 for describing orientation. But we have only ...
2
votes
1answer
94 views

Is the energy per degree of freedom $\frac{1}{2}kT$ in relativistic systems?

The equipartition theorem says that the mean energy per degree of freedom is $\frac{1}{2} kT$. Is this result relativistically correct?
2
votes
1answer
65 views

Degrees of freedom of a point mass sliding on a rigid curved wire without friction

I am very new to the subject and am going through Structure and Interpretation of Classical Mechanics. One exercise asks to find the degrees of freedom of a number of systems, one of which is a ...
2
votes
1answer
594 views

Degrees of freedom in Quantum Mechanics

If we look at a particle in classical mechanics, the degrees of freedom increase as its size decreases like the degrees of freedom of an atom is more than that of molecule, and subsequently, the ...
2
votes
1answer
918 views

How come a rigid body has 6 degrees of freedoms (DOFs) ? Isn't velocity a DOF?

For rigid body we need to know position of three points and their velocities to determine everything. So that would make 12 DOF. Why do text books say it has six DOFs?
2
votes
1answer
516 views

Degrees of freedom in a diatomic molecule [duplicate]

We know that a monatomic compound can only have 3 degrees of freedom as we can consider it to be a point mass. However now that we consider a diatomic molecule, there are 3 degrees of freedom in ...
2
votes
1answer
163 views

Why does the Dirac equation reduce the fermionic degree of freedom by half

We know that in 4D a Dirac spinor has 4 complex components or 8 real components meaning 8 real off shell degrees of freedom (please correct me if I say something wrong here). When we go on-shell i.e ...
2
votes
2answers
338 views

First-order and second-order wave equations, versus the uncertainty principle

In classical physics, we have second-order equations like Newton's laws, so we need to specify both position (zeroth order) and velocity (first order) of a particle as initial conditions, in order to ...
2
votes
1answer
121 views

What are “local degrees of freedom in gravity”, and why do they lead to fixed energy densities?

I am reading Jan de Boer's review of the AdS/CFT correspondence and I quote from end of page 1, where he is talking about equivalence of $(d+1)$-dimensional gravity to $d$-dimensional field theory ...
2
votes
1answer
418 views

Degrees of freedom in the infinite momentum frame

Lenny Susskind explains in this video at about 40min, as an extended object (for example a relativistic string) is boosted to the infinite momentum frame (sometimes called light cone frame), it has no ...
2
votes
1answer
479 views

Violations of Dulong-Petit rule as an upper limit to heat capacity

Does any known substance have a heat capacity at constant volume ($C_V$) per mole of atoms greater than $3k_B$ ~ 24.94 J/(mol K)? In order to count, the substance must actually be made of atoms, that ...